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Definition/Summary
The dicyclic group or generalized quaternion group Dic(n) is a nonabelian group with order 4n that is related to the cyclic group Z(2n).
It is closely related to the dihedral group.
Equations
It has two generators, a and b, which satisfy
[itex]a^{2n} = e ,\ b^2 = a^n ,\ bab^{-1} = a^{-1}[/itex]
Its elements are
[itex]Dic_n = \{a^k, ba^k : 0 \leq k < 2n \}[/itex]
Its "reflection-like" elements all have order 4, unlike the similar elements of the dihedral group with order 2.
[itex](ba^k)^4 = e[/itex]
Extended explanation
This group may be realized as the matrices
[itex]a^k = \begin{pmatrix} \cos\theta_k & - \sin\theta_k \\ \sin\theta_k & \cos\theta_k \end{pmatrix}[/itex]
[itex]ba^k = \begin{pmatrix} i \cos\theta_k & - i \sin\theta_k \\ - i \sin\theta_k & - i \cos\theta_k \end{pmatrix}[/itex]
where
[itex]\theta_k = \frac{\pi k}{n}[/itex]
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
The dicyclic group or generalized quaternion group Dic(n) is a nonabelian group with order 4n that is related to the cyclic group Z(2n).
It is closely related to the dihedral group.
Equations
It has two generators, a and b, which satisfy
[itex]a^{2n} = e ,\ b^2 = a^n ,\ bab^{-1} = a^{-1}[/itex]
Its elements are
[itex]Dic_n = \{a^k, ba^k : 0 \leq k < 2n \}[/itex]
Its "reflection-like" elements all have order 4, unlike the similar elements of the dihedral group with order 2.
[itex](ba^k)^4 = e[/itex]
Extended explanation
This group may be realized as the matrices
[itex]a^k = \begin{pmatrix} \cos\theta_k & - \sin\theta_k \\ \sin\theta_k & \cos\theta_k \end{pmatrix}[/itex]
[itex]ba^k = \begin{pmatrix} i \cos\theta_k & - i \sin\theta_k \\ - i \sin\theta_k & - i \cos\theta_k \end{pmatrix}[/itex]
where
[itex]\theta_k = \frac{\pi k}{n}[/itex]
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!